Bipartite random friends-and-strangers graph component conjecture

Let $r$ be a positive integer, let $p\in[0,1]$, and let $X$ and $Y$ be independently chosen random graphs from $\mathcal G(K_{r,r},p)$, where $K_{r,r}$ is the complete bipartite graph with parts of size $r$. Write $\operatorname{\mathsf{FS}}(X,Y)$ for their friends-and-strangers graph. Bipartite component conjecture. There exists an absolute constant $C>0$ such that if $p\geq Cr^{-1/2}$, then $\operatorname{\mathsf{FS}}(X,Y)$ has exactly $2$ connected components with high probability. This conjecture concerns the typical number of components in the bipartite random setting and would sharpen the paper’s estimates; its resolution is not supplied in the source.